How Do You Calculate Total Energy and Potential Energy in SHM? (HL)

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The Energy Mechanism

In simple harmonic motion (SHM), acceleration is proportional to displacement and acts towards equilibrium:

Multiplying by mass gives the restoring force:

This has the same form as Hooke's law, $F=-kx$, so $k=m\omega^2$. The potential energy stored at displacement $x$ is therefore

At either maximum displacement, $x=\pm x_0$, the instantaneous speed is zero. All the system's mechanical energy is then potential energy, giving the constant total energy

The kinetic energy at any displacement follows from conservation of energy:

For ideal SHM without resistive forces, energy continuously transfers between potential and kinetic forms while total energy remains constant.

The equations also show how energy depends on amplitude and displacement. Doubling the amplitude makes the total energy four times larger because $E_T$ is proportional to $x_0^2$. At a displacement equal to half the amplitude, $E_p/E_T=x^2/x_0^2=1/4$, so one quarter of the energy is potential and three quarters is kinetic. The sign of $x$ does not affect either energy because displacement is squared. This means equal positive and negative displacements have identical energy distributions, although their velocities point in opposite directions.

Worked Example

For $m=0.40\,\text{kg}$, $\omega=5.0\,\text{rad s}^{-1}$, $x_0=0.12\,\text{m}$, and $x=0.072\,\text{m}$:

Therefore,

The three energies satisfy $E_T=E_p+E_k$, confirming conservation of mechanical energy.

Exam Technique

These quantitative equations are Additional Higher Level (AHL) content in C.1. In a calculation, state the equation, substitute SI values, and give the energy in joules.

A common misconception is that $x$ and $x_0$ are interchangeable. Use $x_0$ only for total energy and the current displacement $x$ for potential energy. Also avoid claiming that total energy varies during ideal SHM; only its kinetic and potential components change.